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Quantifier Elimination via Functional Composition

Jie-Hong Roland Jiang

Dept. of Electrical Eng. / Grad. Inst. of Electronics Eng.National Taiwan UniversityTaipei 10617, Taiwan

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Outline

Motivations

Prior work

Quantifier elimination by functional composition Propositional logic Predicate logic

Experimental results

Conclusions

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Introduction

Quantifier elimination transforms a quantified formula, e.g., x1x2x3 xn , into an equivalent quantifier-free formula can be preferable to x1x2x3 xn

E.g., Properties of can be reasoned more easily can be treated as a synthesis result for implementation

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Introduction

QE examplesGauss elimination for systems of linear equ

alities

Fourier-Motzkin elimination for systems of linear inequalities

Cylindrical algebraic decomposition for systems of polynomial inequalities

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Motivations

QE arises in many contexts, including computation theory, mathematical logic, optimization, …Constraint reductionQuantified Boolean Formula (QBF) solving

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Main focus

Propositional logicQuantifier elimination for QBFs

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Prior work

Formula expansion y (x,y) = (x,0) (x,1) BDD, AIG based image-computation [Coudert90][Pigorsch06]

Normal-form conversion Existential (universal) quantification is computationally trivial for disjunctive (c

onjunctive) normal form formulas Simply remove from the formula the literals of variables to be quantified

E.g., x1[(x1 x2 x3)(x1 x3)(x2 x4)] = (x2 x3)(x3)(x2 x4) Formula conversion between CNF and DNF [McMillan02]

Solution enumeration Compute (x) = y (x,y) by enumerating all satisfiable assignments on x SAT-based image computation, e.g., [Ganai04]

Yet another way?

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Question

Given a quantified formula y (x,y), what should a function f be such that (x,f(x)) = y (x,y)?

I.e., QE by functional composition

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Answer

(x,f(x)) = y (x,y) if and only if f has

care onset (x,1) (x,0) care offset (x,0) (x,1) don’t care set (x,1) (x,0)

In other words,((x,1) (x,0)) f ((x,0) (x,1))

Such f always exists

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Problem formulation

For universal quantification y (x,y) = y (x,y) = (x,f(x)) = (x,f(x)) f has

care onset (x,1) (x,0) care offset (x,0) (x,1) don’t care set (x,1) (x,0)

So by computing composite functions f, one can iteratively eliminate the quantifiers of any QBF

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Computation

f can be computed byBinary decision diagrams (BDDs)

Not scalable for large Craig interpolation

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Craig interpolation

(Propositional logic)

For A B unsatisfiable, there exists an interpolant of A w.r.t. B such that

1. A 2. B is unsatisfiable

3. refers only to the common variables of A and B

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Computation

A

care onsetB

care offset

interpolant

The interpolant is a valid implementation of f, which can be obtained from the refutation of A B in SAT solving and can be naturally represented in And-Inverter Graphs (AIGs)

care onset (x,1) (x,0) care offset (x,0) (x,1) don’t care set (x,1) (x,0)

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Composition vs. expansion

Is (x,f(x)) better than (x,0) (x,1) ?

f

x x

in terms of AIGs, where structurally identical nodes are merged

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Composition vs. expansion

[] Consider simplifying (x,1) in (x,0) (x,1) using (x,0) as don’t carecare onset (x,1) (x,0) care offset (x,1) (x,0)

In contrast to f withcare onset (x,1) (x,0) care offset (x,0) (x,1)

For existential quantification, composition can be much better than expansion for sparse (due to simple interpolants)

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Composition vs. expansion

[] Consider simplifying (x,1) in (x,0) (x,1) using (x,0) as don’t carecare onset (x,1) (x,0) care offset (x,1) (x,0)

In contrast to f withcare onset (x,1) (x,0) care offset (x,0) (x,1)

For universal quantification, composition can be much better than expansion for dense (due to simple interpolants)

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Generalization to predicate logic

For a language L in predicate logic under structure (interpretation) I, |=I x(y (x,y) = F (x,Fx))

QE is possible if such function F is finitely expressible in the language

If y (x,y) = (x,fx), then (a,b)y (a,y) is satisfied for any a, b with f(a)=b

If for any a, b with f(a)=b satisfies (a,b)y (a,y), then

y (x,y) = (i (x,fix)), where f = fi if i holds {(a,b) | (a,b)y (a,y)} characterizes the flexibility of f,

which can be exploited to simplify QE

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Generalization to predicate logic

Examplex(ax2+c=0) over the real number

f(a,c) = (–c/a)1/2 if c/a 0 – if c/a > 0

Taking f(a,c) = (((–c/a)2)1/2)1/2, this quantified formula is equivalent to a((((–c/a)2)1/2)1/2)2+c=0

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Experiments

Given a sequential circuit, we compute its transition relation with input variables being quantified out, i.e.,

x [i (si' i(x,s))]

Simple quantification scheduling appliedAIG minimization applied

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Experimental results

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Discussion

Expansion vs. composition based QEAnalogy with two-level vs. multi-level circuit

minimizationRelaxing level constraints admits more compact

circuit representation

Sparsity may play an essential role in the effectiveness of composition-based QE

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Conclusions

Quantifier elimination with functional composition can be effective at least for some applications (where the sparsity condition holds)

Future workFind more applicationsQE in predicate logic

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Thanks for your attention!

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Questions?